Asymptotic Limit Of Viscous Shallow Water Equations

This paper mainly researchs the inviscid and low Froude number limit of viscous shallow water and Euler equations in two dimensional whole spaceR~2,and the low Froude limit of the rotating shallow water and Euler system in the two dimensional bounded areas Ω.The full text is divided into five chapters.The first chapter introduces the research background,significance and existing research results of shallow water equations,and gives the research goals and directions of this article.In the second chapter,the common vector formulas and inequalities are given.In chapter 3,the limits of inviscosity and low Froude number for viscous shallow water and Euler equations are studied.By means of the classical energy method and the relative entropy method,it is proved that the classical solution of the viscous shallow water equations converges to the classical solution of the incompressible Euler equations under the general initial value condition when the viscosity coefficient and Froude number approach zero simultaneously,and the convergence rate is given.In chapter 4,the low Froude numbers of rotating shallow water and Euler equations are studied.Based on the measure solution of rotating shallow water system,the relative entropy method is used to prove that the measure solution of rotating shallow water system converges to the classical solution of rotating lake system under appropriate initial conditions.The last chapter summarizes the main results of this paper.Based on the existing results,the future research direction is proposed.